Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.2% → 99.2%

Time: 9.1s

Alternatives: 24

Speedup: 18.5×

Specification

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * cos(((single(2.0) * single(pi)) * u2));
end

Initial Program: 57.2% accurate, 1.0× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * cos(((single(2.0) * single(pi)) * u2));
end

Alternative 1: 99.2% accurate, 0.9× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left(-2 \cdot u2, \pi, 1.5707963705062866\right)\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (*
 (sqrt (- (log1p (- u1))))
 (sin (fma (* -2.0 u2) PI 1.5707963705062866))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf(fmaf((-2.0f * u2), ((float) M_PI), 1.5707963705062866f));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(fma(Float32(Float32(-2.0) * u2), Float32(pi), Float32(1.5707963705062866))))
end

Derivation

1. Initial program 57.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

3. Applied rewrites 57.2%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(-2 \cdot u2, \pi, 0.5 \cdot \pi\right)\right) \]

4. Evaluated real constant 57.2%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\mathsf{fma}\left(-2 \cdot u2, \pi, 1.5707963705062866\right)\right) \]
5. Step-by-step derivation

6. Applied rewrites 99.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left(-2 \cdot u2, \pi, 1.5707963705062866\right)\right) \]
7. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log1p (- u1)))) (cos (* 6.2831854820251465 u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * cosf((6.2831854820251465f * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * cos(Float32(Float32(6.2831854820251465) * u2)))
end

Derivation

1. Initial program 57.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

2. Evaluated real constant 57.2%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]
3. Step-by-step derivation

4. Applied rewrites 99.0%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]
5. Add Preprocessing

Alternative 3: 96.7% accurate, 0.9× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.05999999865889549:\\ \;\;\;\;t\_0 + -19.739209900765786 \cdot \left({u2}^{2} \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (let* ((t_0 (sqrt (- (log1p (- u1))))))
  (if (<= (* (* 2.0 PI) u2) 0.05999999865889549)
    (+ t_0 (* -19.739209900765786 (* (pow u2 2.0) t_0)))
    (*
     (sqrt (fma (* 0.5 u1) u1 u1))
     (cos (* 6.2831854820251465 u2))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-log1pf(-u1));
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.05999999865889549f) {
		tmp = t_0 + (-19.739209900765786f * (powf(u2, 2.0f) * t_0));
	} else {
		tmp = sqrtf(fmaf((0.5f * u1), u1, u1)) * cosf((6.2831854820251465f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log1p(Float32(-u1))))
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.05999999865889549))
		tmp = Float32(t_0 + Float32(Float32(-19.739209900765786) * Float32((u2 ^ Float32(2.0)) * t_0)));
	else
		tmp = Float32(sqrt(fma(Float32(Float32(0.5) * u1), u1, u1)) * cos(Float32(Float32(6.2831854820251465) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0599999987

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Evaluated real constant 57.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + \frac{-173627926472025}{8796093022208} \cdot \left({u2}^{2} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 52.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} + -19.739209900765786 \cdot \left({u2}^{2} \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 88.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} + -19.739209900765786 \cdot \left({u2}^{2} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \]

   if 0.0599999987 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 88.3%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 88.3%

      \[\leadsto \sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   7. Evaluated real constant 88.3%

      \[\leadsto \sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 96.6% accurate, 0.9× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.05999999865889549:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<= (* (* 2.0 PI) u2) 0.05999999865889549)
  (*
   (sqrt (- (log1p (- u1))))
   (fma (* -19.739209900765786 u2) u2 1.0))
  (* (sqrt (fma (* 0.5 u1) u1 u1)) (cos (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.05999999865889549f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
	} else {
		tmp = sqrtf(fmaf((0.5f * u1), u1, u1)) * cosf((6.2831854820251465f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.05999999865889549))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	else
		tmp = Float32(sqrt(fma(Float32(Float32(0.5) * u1), u1, u1)) * cos(Float32(Float32(6.2831854820251465) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0599999987

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Evaluated real constant 57.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 52.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 88.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

   if 0.0599999987 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 88.3%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 88.3%

      \[\leadsto \sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   7. Evaluated real constant 88.3%

      \[\leadsto \sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 96.6% accurate, 0.9× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.05999999865889549:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \cos \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<= (* (* 2.0 PI) u2) 0.05999999865889549)
  (*
   (sqrt (- (log1p (- u1))))
   (fma (* -19.739209900765786 u2) u2 1.0))
  (* (sqrt (* (fma 0.5 u1 1.0) u1)) (cos (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.05999999865889549f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * cosf((6.2831854820251465f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.05999999865889549))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * cos(Float32(Float32(6.2831854820251465) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0599999987

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Evaluated real constant 57.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 52.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 88.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

   if 0.0599999987 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 88.3%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Evaluated real constant 88.3%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 88.3%

      \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 94.5% accurate, 0.9× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.11999999731779099:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\mathsf{fma}\left(\mathsf{fma}\left(-2, u2, 1\right), \pi, \pi\right)\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<= (* (* 2.0 PI) u2) 0.11999999731779099)
  (*
   (sqrt (- (log1p (- u1))))
   (fma (* -19.739209900765786 u2) u2 1.0))
  (* (sqrt u1) (cos (fma (fma -2.0 u2 1.0) PI PI)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.11999999731779099f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
	} else {
		tmp = sqrtf(u1) * cosf(fmaf(fmaf(-2.0f, u2, 1.0f), ((float) M_PI), ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.11999999731779099))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	else
		tmp = Float32(sqrt(u1) * cos(fma(fma(Float32(-2.0), u2, Float32(1.0)), Float32(pi), Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.119999997

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Evaluated real constant 57.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 52.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 88.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

   if 0.119999997 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 76.7%

      \[\leadsto \sqrt{u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 76.6%

      \[\leadsto \sqrt{u1} \cdot \cos \left(\mathsf{fma}\left(-2 \cdot u2, \pi, \pi\right) + \pi\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 76.6%

      \[\leadsto \sqrt{u1} \cdot \cos \left(\mathsf{fma}\left(\mathsf{fma}\left(-2, u2, 1\right), \pi, \pi\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 94.4% accurate, 1.0× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.11999999731779099:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-2, u2, 0.5\right) \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<= (* (* 2.0 PI) u2) 0.11999999731779099)
  (*
   (sqrt (- (log1p (- u1))))
   (fma (* -19.739209900765786 u2) u2 1.0))
  (* (sqrt u1) (sin (* (fma -2.0 u2 0.5) PI)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.11999999731779099f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
	} else {
		tmp = sqrtf(u1) * sinf((fmaf(-2.0f, u2, 0.5f) * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.11999999731779099))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	else
		tmp = Float32(sqrt(u1) * sin(Float32(fma(Float32(-2.0), u2, Float32(0.5)) * Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.119999997

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Evaluated real constant 57.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 52.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 88.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

   if 0.119999997 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 76.7%

      \[\leadsto \sqrt{u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 76.8%

      \[\leadsto \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-2, u2, 0.5\right) \cdot \pi\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 94.4% accurate, 1.0× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.11999999731779099:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(\left|u2\right|, 6.2831854820251465, 1.5707963705062866\right)\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<= (* (* 2.0 PI) u2) 0.11999999731779099)
  (*
   (sqrt (- (log1p (- u1))))
   (fma (* -19.739209900765786 u2) u2 1.0))
  (*
   (sqrt u1)
   (sin (fma (fabs u2) 6.2831854820251465 1.5707963705062866)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.11999999731779099f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
	} else {
		tmp = sqrtf(u1) * sinf(fmaf(fabsf(u2), 6.2831854820251465f, 1.5707963705062866f));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.11999999731779099))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	else
		tmp = Float32(sqrt(u1) * sin(fma(abs(u2), Float32(6.2831854820251465), Float32(1.5707963705062866))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.119999997

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Evaluated real constant 57.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 52.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 88.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

   if 0.119999997 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 76.7%

      \[\leadsto \sqrt{u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Evaluated real constant 76.7%

      \[\leadsto \sqrt{u1} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 76.7%

      \[\leadsto \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(\left|u2\right|, 6.2831854820251465, 0.5 \cdot \pi\right)\right) \]

   8. Evaluated real constant 76.7%

      \[\leadsto \sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(\left|u2\right|, 6.2831854820251465, 1.5707963705062866\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 94.4% accurate, 1.1× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.11999999731779099:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(6.2831854820251465 \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<= (* (* 2.0 PI) u2) 0.11999999731779099)
  (*
   (sqrt (- (log1p (- u1))))
   (fma (* -19.739209900765786 u2) u2 1.0))
  (* (sqrt u1) (cos (* 6.2831854820251465 u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.11999999731779099f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
	} else {
		tmp = sqrtf(u1) * cosf((6.2831854820251465f * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.11999999731779099))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	else
		tmp = Float32(sqrt(u1) * cos(Float32(Float32(6.2831854820251465) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.119999997

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Evaluated real constant 57.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 52.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 88.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 88.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

   if 0.119999997 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 76.7%

      \[\leadsto \sqrt{u1} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Evaluated real constant 76.7%

      \[\leadsto \sqrt{u1} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 88.0% accurate, 1.4× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (*
 (sqrt (- (log1p (- u1))))
 (+ 1.0 (* -19.739209900765786 (pow u2 2.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * (1.0f + (-19.739209900765786f * powf(u2, 2.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(Float32(1.0) + Float32(Float32(-19.739209900765786) * (u2 ^ Float32(2.0)))))
end

Derivation

1. Initial program 57.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

2. Evaluated real constant 57.2%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

3. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
4. Step-by-step derivation

5. Applied rewrites 52.4%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
6. Step-by-step derivation

7. Applied rewrites 88.0%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
8. Add Preprocessing

Alternative 11: 88.0% accurate, 2.1× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log1p (- u1)))) (fma (* -19.739209900765786 u2) u2 1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)))
end

Derivation

1. Initial program 57.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

2. Evaluated real constant 57.2%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

3. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
4. Step-by-step derivation

5. Applied rewrites 52.4%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
6. Step-by-step derivation

7. Applied rewrites 88.0%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
8. Step-by-step derivation

9. Applied rewrites 88.0%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]
10. Add Preprocessing

Alternative 12: 86.6% accurate, 0.7× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(1 + \left(-19.739209900765786 \cdot u2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(-19.739209900765786, u2 \cdot u2, 1\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
  (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.05000000074505806)
    (*
     (sqrt (* u1 (+ 1.0 (* 0.5 u1))))
     (+ 1.0 (* (* -19.739209900765786 u2) u2)))
    (* t_0 (fma -19.739209900765786 (* u2 u2) 1.0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if ((t_0 * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.05000000074505806f) {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * (1.0f + ((-19.739209900765786f * u2) * u2));
	} else {
		tmp = t_0 * fmaf(-19.739209900765786f, (u2 * u2), 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (Float32(t_0 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.05000000074505806))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * Float32(Float32(1.0) + Float32(Float32(Float32(-19.739209900765786) * u2) * u2)));
	else
		tmp = Float32(t_0 * fma(Float32(-19.739209900765786), Float32(u2 * u2), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0500000007

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Evaluated real constant 57.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 52.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 79.1%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 79.1%

       \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(1 + \left(-19.739209900765786 \cdot u2\right) \cdot u2\right) \]

   if 0.0500000007 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Evaluated real constant 57.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 52.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 52.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786, u2 \cdot u2, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 85.4% accurate, 0.7× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(1 + \left(-19.739209900765786 \cdot u2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<=
     (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2)))
     0.05000000074505806)
  (*
   (sqrt (* u1 (+ 1.0 (* 0.5 u1))))
   (+ 1.0 (* (* -19.739209900765786 u2) u2)))
  (sqrt (- (log1p (- u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.05000000074505806f) {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * (1.0f + ((-19.739209900765786f * u2) * u2));
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.05000000074505806))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * Float32(Float32(1.0) + Float32(Float32(Float32(-19.739209900765786) * u2) * u2)));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0500000007

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Evaluated real constant 57.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 52.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 79.1%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 79.1%

       \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(1 + \left(-19.739209900765786 \cdot u2\right) \cdot u2\right) \]

   if 0.0500000007 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.8%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 79.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 85.4% accurate, 0.7× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.05000000074505806:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<=
     (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2)))
     0.05000000074505806)
  (*
   (sqrt (* u1 (+ 1.0 (* 0.5 u1))))
   (fma (* -19.739209900765786 u2) u2 1.0))
  (sqrt (- (log1p (- u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.05000000074505806f) {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.05000000074505806))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0500000007

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Evaluated real constant 57.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 52.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 79.1%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 79.1%

       \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

   if 0.0500000007 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.8%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 79.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 85.4% accurate, 0.7× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.05000000074505806:\\ \;\;\;\;\mathsf{fma}\left(-19.739209900765786, u2 \cdot u2, 1\right) \cdot \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<=
     (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2)))
     0.05000000074505806)
  (*
   (fma -19.739209900765786 (* u2 u2) 1.0)
   (sqrt (* (fma 0.5 u1 1.0) u1)))
  (sqrt (- (log1p (- u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.05000000074505806f) {
		tmp = fmaf(-19.739209900765786f, (u2 * u2), 1.0f) * sqrtf((fmaf(0.5f, u1, 1.0f) * u1));
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.05000000074505806))
		tmp = Float32(fma(Float32(-19.739209900765786), Float32(u2 * u2), Float32(1.0)) * sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0500000007

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Evaluated real constant 57.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 52.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 79.1%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 79.1%

       \[\leadsto \mathsf{fma}\left(-19.739209900765786, u2 \cdot u2, 1\right) \cdot \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \]

   if 0.0500000007 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.8%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 79.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 83.0% accurate, 2.1× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.017999999225139618:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} - 19.739209900765786 \cdot \left(\left(u2 \cdot u2\right) \cdot \sqrt{u1}\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<= (* (* 2.0 PI) u2) 0.017999999225139618)
  (sqrt (- (log1p (- u1))))
  (- (sqrt u1) (* 19.739209900765786 (* (* u2 u2) (sqrt u1))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.017999999225139618f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = sqrtf(u1) - (19.739209900765786f * ((u2 * u2) * sqrtf(u1)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.017999999225139618))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(sqrt(u1) - Float32(Float32(19.739209900765786) * Float32(Float32(u2 * u2) * sqrt(u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0179999992

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.8%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 79.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]

   if 0.0179999992 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Evaluated real constant 57.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + \frac{-173627926472025}{8796093022208} \cdot \left({u2}^{2} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 52.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} + -19.739209900765786 \cdot \left({u2}^{2} \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} + -19.739209900765786 \cdot \left({u2}^{2} \cdot \sqrt{u1}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.5%

      \[\leadsto \sqrt{u1} + -19.739209900765786 \cdot \left({u2}^{2} \cdot \sqrt{u1}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 69.5%

       \[\leadsto \sqrt{u1} - 19.739209900765786 \cdot \left(\left(u2 \cdot u2\right) \cdot \sqrt{u1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 83.0% accurate, 2.2× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.017999999225139618:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \sqrt{u1}, -19.739209900765786, \sqrt{u1}\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<= (* (* 2.0 PI) u2) 0.017999999225139618)
  (sqrt (- (log1p (- u1))))
  (fma (* (* u2 u2) (sqrt u1)) -19.739209900765786 (sqrt u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.017999999225139618f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = fmaf(((u2 * u2) * sqrtf(u1)), -19.739209900765786f, sqrtf(u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.017999999225139618))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = fma(Float32(Float32(u2 * u2) * sqrt(u1)), Float32(-19.739209900765786), sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0179999992

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.8%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 79.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]

   if 0.0179999992 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Evaluated real constant 57.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} + \frac{-173627926472025}{8796093022208} \cdot \left({u2}^{2} \cdot \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 52.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} + -19.739209900765786 \cdot \left({u2}^{2} \cdot \sqrt{-\log \left(1 - u1\right)}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 52.4%

      \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}, -19.739209900765786, \sqrt{-\log \left(1 - u1\right)}\right) \]

   8. Taylor expanded in u1 around 0

      \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \sqrt{u1}, -19.739209900765786, \sqrt{u1}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 69.5%

       \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \sqrt{u1}, -19.739209900765786, \sqrt{u1}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 83.0% accurate, 2.2× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.017999999225139618:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + \left(-19.739209900765786 \cdot u2\right) \cdot u2\right)\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (if (<= (* (* 2.0 PI) u2) 0.017999999225139618)
  (sqrt (- (log1p (- u1))))
  (* (sqrt u1) (+ 1.0 (* (* -19.739209900765786 u2) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.017999999225139618f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = sqrtf(u1) * (1.0f + ((-19.739209900765786f * u2) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.017999999225139618))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(sqrt(u1) * Float32(Float32(1.0) + Float32(Float32(Float32(-19.739209900765786) * u2) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0179999992

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.8%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \]
   5. Step-by-step derivation

   6. Applied rewrites 79.4%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]

   if 0.0179999992 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Evaluated real constant 57.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 52.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.5%

      \[\leadsto \sqrt{u1} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 69.5%

       \[\leadsto \sqrt{u1} \cdot \left(1 + \left(-19.739209900765786 \cdot u2\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 79.4% accurate, 0.8× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.025200000032782555:\\ \;\;\;\;\sqrt{u1} \cdot \left(1 + \left(-19.739209900765786 \cdot u2\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
  (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.025200000032782555)
    (* (sqrt u1) (+ 1.0 (* (* -19.739209900765786 u2) u2)))
    t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if ((t_0 * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.025200000032782555f) {
		tmp = sqrtf(u1) * (1.0f + ((-19.739209900765786f * u2) * u2));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (Float32(t_0 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.025200000032782555))
		tmp = Float32(sqrt(u1) * Float32(Float32(1.0) + Float32(Float32(Float32(-19.739209900765786) * u2) * u2)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = sqrt(-log((single(1.0) - u1)));
	tmp = single(0.0);
	if ((t_0 * cos(((single(2.0) * single(pi)) * u2))) <= single(0.025200000032782555))
		tmp = sqrt(u1) * (single(1.0) + ((single(-19.739209900765786) * u2) * u2));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0252

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Evaluated real constant 57.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 52.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.5%

      \[\leadsto \sqrt{u1} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 69.5%

       \[\leadsto \sqrt{u1} \cdot \left(1 + \left(-19.739209900765786 \cdot u2\right) \cdot u2\right) \]

   if 0.0252 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.8%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 79.4% accurate, 0.8× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.025200000032782555:\\ \;\;\;\;\sqrt{u1} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
  (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.025200000032782555)
    (* (sqrt u1) (fma (* -19.739209900765786 u2) u2 1.0))
    t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if ((t_0 * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.025200000032782555f) {
		tmp = sqrtf(u1) * fmaf((-19.739209900765786f * u2), u2, 1.0f);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (Float32(t_0 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.025200000032782555))
		tmp = Float32(sqrt(u1) * fma(Float32(Float32(-19.739209900765786) * u2), u2, Float32(1.0)));
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0252

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Evaluated real constant 57.2%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(6.2831854820251465 \cdot u2\right) \]

   3. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + \frac{-173627926472025}{8796093022208} \cdot {u2}^{2}\right) \]
   4. Step-by-step derivation

   5. Applied rewrites 52.4%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]

   6. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.5%

      \[\leadsto \sqrt{u1} \cdot \left(1 + -19.739209900765786 \cdot {u2}^{2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 69.5%

       \[\leadsto \sqrt{u1} \cdot \mathsf{fma}\left(-19.739209900765786 \cdot u2, u2, 1\right) \]

   if 0.0252 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.8%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 78.4% accurate, 0.8× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.07999999821186066:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
  (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.07999999821186066)
    (sqrt (fma (* 0.5 u1) u1 u1))
    t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if ((t_0 * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.07999999821186066f) {
		tmp = sqrtf(fmaf((0.5f * u1), u1, u1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (Float32(t_0 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.07999999821186066))
		tmp = sqrt(fma(Float32(Float32(0.5) * u1), u1, u1));
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0799999982

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.8%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \]

   5. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.4%

      \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 72.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)} \]

   if 0.0799999982 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 57.2%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 48.8%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 72.4% accurate, 5.1× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (sqrt (fma (* 0.5 u1) u1 u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(fmaf((0.5f * u1), u1, u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(fma(Float32(Float32(0.5) * u1), u1, u1))
end

Derivation

1. Initial program 57.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \]
3. Step-by-step derivation

4. Applied rewrites 48.8%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \]

5. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 72.4%

   \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \]
8. Step-by-step derivation

9. Applied rewrites 72.4%

   \[\leadsto \sqrt{\mathsf{fma}\left(0.5 \cdot u1, u1, u1\right)} \]
10. Add Preprocessing

Alternative 23: 72.4% accurate, 5.1× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (sqrt (* (fma 0.5 u1 1.0) u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((fmaf(0.5f, u1, 1.0f) * u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1))
end

Derivation

1. Initial program 57.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \]
3. Step-by-step derivation

4. Applied rewrites 48.8%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \]

5. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)} \]
6. Step-by-step derivation

7. Applied rewrites 72.4%

   \[\leadsto \sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \]
8. Step-by-step derivation

9. Applied rewrites 72.4%

   \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \]
10. Add Preprocessing

Alternative 24: 64.7% accurate, 18.5× speedup

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\sqrt{u1} \]

FPCore

(FPCore (cosTheta_i u1 u2)
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0))
          (and (<= 2.328306437e-10 u1) (<= u1 1.0)))
     (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (sqrt u1))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1);
}

Fortran

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1);
end

Derivation

1. Initial program 57.2%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

2. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\mathsf{neg}\left(\log \left(1 - u1\right)\right)} \]
3. Step-by-step derivation

4. Applied rewrites 48.8%

   \[\leadsto \sqrt{-\log \left(1 - u1\right)} \]

5. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{u1} \]
6. Step-by-step derivation

7. Applied rewrites 64.7%

   \[\leadsto \sqrt{u1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2026070 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))